1. Field of the Invention
The present invention relates to communications systems that use digital filters. More specifically, the invention relates to adaptation circuits and methods for adjusting the filter coefficients of a transversal or other digital filter.
2. Description of the Related Art
Filtering is a common and powerful function that finds use in a large variety of applications. One important application is communications, in which information is sent from one place to another over a channel. A filter used to compensate for the effects of the channel is commonly referred to as an equalizer.
A major source of error in information transmission is inter-symbol interference (ISI), which arises when a signal is sent across a dispersive channel. Dispersive channels tend to spread the energy of a transmitted signal out over time, which means both past and future symbols can interfere with the current symbol.
To further illustrate this point, consider a transmitted signal, x[k], which is sent across a dispersive channel with impulse response h[k]. The received signal, y[k], is given by:
                                                                        y                ⁡                                  [                  k                  ]                                            =                            ⁢                                                ∑                  n                                ⁢                                                      h                    ⁡                                          [                      n                      ]                                                        ⁢                                      x                    ⁡                                          [                                              k                        -                        n                                            ]                                                                                                                                              =                            ⁢                                                                    h                    ⁡                                          [                      0                      ]                                                        ⁢                                      x                    ⁡                                          [                      k                      ]                                                                      +                                                      ∑                                          n                      <                      0                                                        ⁢                                                            h                      ⁡                                              [                        n                        ]                                                              ⁢                                          x                      ⁡                                              [                                                  k                          -                          n                                                ]                                                                                            +                                                      ∑                                          n                      >                      0                                                        ⁢                                                            h                      ⁡                                              [                        n                        ]                                                              ⁢                                          x                      ⁡                                              [                                                  k                          -                          n                                                ]                                                                                                                                                    Eqn        ⁢                                  ⁢        1            
The second term in equation 1 arises from the precursor component of the channel impulse response, and allows future symbols to interfere with the current symbol. The third term in equation 1 arises from the postcursor component of the channel impulse response, and allows previous symbols to interfere with the current symbol. Fortunately, equalization can be used to reduce or remove these components.
Oftentimes, one has no prior knowledge of the channel characteristics, making it difficult define the optimum filter. To overcome this problem, filters are often made adaptive, allowing them to “learn” the channel characteristics.
Adaptive Transversal Filters
The adaptive transversal filter is a fundamental component in adaptive equalization applications, and is a well understood non-recursive structure. Adaptive traversal filters commonly operate in the discrete time domain and have a finite impulse response (FIR). A generalized block diagram of a typical adaptive transversal filter 20 is shown in FIG. 1. For convenience, the input history and coefficients are expressed as vectors:Xk=[x[k] x[k−1] . . . x[k−N]]T   Eqn 2Wk=[Wo[k] W1[k] . . . WN[k]]T  Eqn 3
Coefficient adaptation is performed based on the desired response, d[k], and the filter output, y[k]. The desired response, d[k], is often a training (pilot) signal, which is essentially a copy of the transmitted sequence stored in the receiver, or the hard decisions of a Decision Feedback Equalizer (DFE). Commonly used adaptation algorithms attempt to minimize the mean-square error, E[εk2], where the error signal is given by:εk=d[k]−y[k]=d[k]−WkTXk  Eqn 4
Expanding the square of the error signal gives:
                                                                        ɛ                k                2                            =                                                (                                                            d                      ⁡                                              [                        k                        ]                                                              -                                                                  W                        k                        T                                            ⁢                                              X                        k                                                                              )                                2                                                                                        =                                                                    d                    ⁡                                          [                      k                      ]                                                        2                                +                                                      W                    k                    T                                    ⁢                                      X                    k                                    ⁢                                      X                    k                    T                                    ⁢                                      W                    k                                                  -                                  2                  ⁢                                      d                    ⁡                                          [                      k                      ]                                                        ⁢                                      X                    k                    T                                    ⁢                                      W                    k                                                                                                          Eqn        ⁢                                  ⁢        5            
To produce a reasonably simplified expression for the mean-square error, the following assumptions may be made: (1) Wk is fixed, and (2) Xk, d[k], and εk are statistically wide-sense stationary. With these assumptions, the mean-square error reduces to:E[εk2]=E[d[k]2]+WTE[XkXkT]W−2E[d[k]XkT]W  Eqn 6
The above equation reveals that the mean-square error is a quadratic function of the coefficient vector W. This quadratic function is referred to as the error surface, and it contains a global minimum at the optimal coefficient vector. The adaptation engine 22 attempts to “walk” the coefficients down the error surface to a point as close as possible to the optimal solution.
A variety of basic algorithms are available to converge the coefficient vector to the optimal solution, including but not limited to Newton's method, the steepest descent method, least-mean square (LMS), and recursive least squares (RLS). LMS is one of the most commonly used algorithms due to its ease of computation. The LMS algorithm achieves its simplicity by approximating the mean-square error, E[εk2], with εk2, leading to the following coefficient update equation:Wk+1=Wk+μεkXk   Eqn 7
In the above equation, μ is a step-size scalar that can be used to control convergence rate and steady-state accuracy.
In the above exemplary description, the filter and associated algorithms operate on real-valued data. The extension to complex-valued data and coefficients is well known in the art and is included in the scope of the present disclosure. Similarly, in the exemplary description the optimal coefficient vector is chosen as the one that minimizes the mean square error between the filter output and the desired response.
Those skilled in the art will recognize that it may be advantageous to choose the optimal coefficient vector based on some criterion other than mean-square error. This results in forms of the coefficient updating equation that may differ significantly from equation 7. The present disclosure is not intended to limit the applicable scope of the invention to the exemplary description and is intended to include such techniques as may be utilized by those skilled in the art.
Local Minima on the Error Surface
Under certain circumstances, local minima can also exist on the error surface. Adaptation engines that become trapped on a local minimum provide a non-optimal coefficient vector, which reduces the effectiveness of the transversal filter. Local minima are typically caused by non-linear effects in the signal path, certain channel characteristics, or some combination of the two.
Blind Equalization
When the desired response, d[k], is unknown, adaptation may be performed in a blind mode. There are many algorithms capable of blindly converging an adaptive filter using higher-order statistics of the filter's input. Some prominent algorithms include Sato's algorithm and the Constant Modulus Algorithm (CMA).
Decision Feedback Equalizers
An alternative to the feedforward transversal filter, known as the Decision Feedback Equalizer (DFE), was originally proposed in 1967 and showed superior performance to its linear counterpart. DFEs were later modified to be adaptive. Adaptive DFEs typically use adaptive transversal filters 20 in both feedforward and feedback roles (although the feedforward transversal filter 20 is commonly omitted), as shown in FIG. 2.
The role of the feedforward section is to reduce the precursor component of the inter-symbol interference, while the feedback section reduces the post-cursor component. Traditional symbol-rate DFEs typically correct for precursor and post-cursor components spaced at integer multiples of the symbol period, T. For example, a DFE with N feedback taps can correct for post-cursor components that occur at intervals of T 2T, . . . , NT from the current symbol.
DFEs can be implemented in analog or digital form. Digital implementations use analog-to-digital conversion circuitry to convert the filter's input signal to digital form.
DFEs are often operated in a decision-directed mode, which uses the output of a decision device as the desired signal. In this case, the error signal is given by the difference between the decision device's output and input. This is advantageous, as it does not require a training signal to converge the adaptation engine, although convergence is inherently more difficult.
A block diagram for a decision-directed DFE is shown in FIG. 3. A common error signal and adaptation engine 22 are used to adapt both feedforward and feedback sections. The generation of this error signal can be challenging, as it is generally necessary to sample and hold, and then scale, the soft decisions (input of the decision device) before subtracting them from the hard decisions (output of the decision device). This allows the delay through the decision device to be accounted for, and also prevents the hard decisions from swamping the small signal level of the soft decisions.
3Fractionally Spaced Equalizers
Fractionally Spaced Equalizers (FSEs) are transversal equalizers (used as a linear equalizer or the feedforward portion of a DFE) whose taps are spaced at some fraction of the symbol period. A typical choice is T/2 spacing, which allows correction of both the in-phase instant and the quadrature instant in the channel impulse response.
For an ideal, jitter-free sampling clock, equalization of anything but the ideal in-phase sampling instant provides no improvement in performance. However, when a realistic, jittered clock is considered, the true sampling instant frequently varies from the ideal point. Because of this, there is an advantage to providing equalization across the entire symbol period. Thus in realistic systems, FSEs provide superior performance to symbol-rate equalizers.
Adaptation Engines
Adaptation engines 22 typically update the complete set of coefficients of the adaptive traversal filter 20 continuously. This has the advantage of providing the fastest possible convergence for a given algorithm. However, the number of operations performed by the adaptation engine 22 in a given period of time is generally directly proportional to the number of coefficients, N. For a large number of coefficients, these designs are therefore commonly inefficient and impractical.
Several designs have been proposed in which the coefficients are updated sequentially and/or in groups. Examples of such designs are set forth in the following references:                J. Sonntag, J. Stonick, J. Gorecki, et al., “An Adaptive PAM-4 5 Gb/s Backplane Transceiver in 0.25 um CMOS,” IEEE 2002 Custom Integrated Circuits Conference, 2002, pp. (20-3-1)-(20-3-4).        V. Wolff, R. Gooch, and J. Treichler “Specification and development of an equalizer-demodulator for wideband digital microwave radio signals,” Proc. IEEE Military Communications Conference, NY, October 1988.        S. Douglas, “Adaptive Filters Employing Partial Updates,” IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing., Vol. 44, No. 3, pp. 209-216, March 1997.        U.S. Pat. Nos. 5,517,213 and 5,157,690.        
These designs generally do not provide sufficient flexibility to optimally balance computational demands, steady-state accuracy, and start-up issues. An example of a start-up issue is the failure of an LMS engine to converge to the optimal set of coefficients due to a received signal with very low Signal to Noise Ratio (SNR).